Integral Closure of Noetherian Domains and Intersections of Rees Valuation Rings, (II)

Paula Kemp; Louis J. Ratliff, Jr.; Kishor Shah

doi:10.18311/jims/2017/6133

Authors

Paula Kemp
Louis J. Ratliff, Jr.
Kishor Shah

DOI:

https://doi.org/10.18311/jims/2017/6133

Keywords:

Integral Closure, Local Domain, Rees Valuation Ring, Unique Factorization Domain

Abstract

Let 1 < s₁ < . . . < s_k be integers, and assume that Îº â‰¥ 2 (so s_k â‰¤ 3). Then there exists a local UFD (Unique Factorization Domain) (R,M) such that:

(1) Height(M) = s_k.

(2) R = R' = âˆ©{VI (V,N) € V_j}, where V_j (j = 1, . . . , Îº) is the set of all of the Rees valuation rings (V,N) of the M-primary ideals such that trd((V I N) I (R I M)) = s_j - 1.

(3) With V₁, . . . , V_Îº as in (2), V₁ âˆª . . . V_Îºis a disjoint union of all of the Rees valuation rings of allof the M-primary ideals, and each M-primary ideal has at least one Rees valuation ring in each V_j .

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References

M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Co., Reading, MA 1969.

R. C. Heitmann, Characterization of completions of unique factorization domains, Trans. Amer. Math. Soc. 337 (1993), 379-387.

I. Kaplansky, Commutative Rings, Allyn and Bacon, Boston, 1970.

D. Katz and J. Validashti, Multiplicities and Rees valuations, Collect. Math. 61 (2010), 1-24.

P. Kemp, L. J. Ratli , Jr., and K. Shah, Integral Closure of Noetherian Domains and Inter- sections of Rees Valuation Rings, (I), J. Indian Math. Soc. (to appear).

H. Matsumura, Commutative Algebra, W. A. Benjamin, NY, 1970.

M. Nagata, Local Rings, Interscience, John Wiley, New York, 1962.

D. G. Northcott, Ideal Theory, Cambridge Tracts in Math. No. 42, Cambridge, 1965.

L. J. Ratli , Jr., On quasi-unmixed local domains, the altitude formula, and the chain con- dition for prime ideals (II), Amer. J. Math. 92 (1970), 99-144.

L. J. Ratli , Jr., On prime divisors of the integral closure of a principal ideal, J. Reine Angew. Math. 255 (1972), 210-220.

I. Swanson and C. Huneke, Integral Closure of Ideals, Rings and Modules, Cambridge Univ. Press, Cambridge, 2006.

O. Zariski and P. Samuel, Commutative Algebra, Vol. 2, D. Van Nostrand, New York, 1960.